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Theorem isoml 39220
Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isoml.b 𝐵 = (Base‘𝐾)
isoml.l = (le‘𝐾)
isoml.j = (join‘𝐾)
isoml.m = (meet‘𝐾)
isoml.o = (oc‘𝐾)
Assertion
Ref Expression
isoml (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem isoml
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isoml.b . . . 4 𝐵 = (Base‘𝐾)
31, 2eqtr4di 2793 . . 3 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6907 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 isoml.l . . . . . . 7 = (le‘𝐾)
64, 5eqtr4di 2793 . . . . . 6 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 5159 . . . . 5 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
8 fveq2 6907 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 isoml.j . . . . . . . 8 = (join‘𝐾)
108, 9eqtr4di 2793 . . . . . . 7 (𝑘 = 𝐾 → (join‘𝑘) = )
11 eqidd 2736 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
12 fveq2 6907 . . . . . . . . 9 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13 isoml.m . . . . . . . . 9 = (meet‘𝐾)
1412, 13eqtr4di 2793 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = )
15 eqidd 2736 . . . . . . . 8 (𝑘 = 𝐾𝑦 = 𝑦)
16 fveq2 6907 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 isoml.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17eqtr4di 2793 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
1918fveq1d 6909 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( 𝑥))
2014, 15, 19oveq123d 7452 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ( 𝑥)))
2110, 11, 20oveq123d 7452 . . . . . 6 (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 (𝑦 ( 𝑥))))
2221eqeq2d 2746 . . . . 5 (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 (𝑦 ( 𝑥)))))
237, 22imbi12d 344 . . . 4 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
243, 23raleqbidv 3344 . . 3 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
253, 24raleqbidv 3344 . 2 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
26 df-oml 39161 . 2 OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
2725, 26elrab2 3698 1 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  OLcol 39156  OMLcoml 39157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-oml 39161
This theorem is referenced by:  isomliN  39221  omlol  39222  omllaw  39225
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